The subject matter of the present invention pertains to a numerical algorithm and software program executed in a computer for determining an improved conductivity profile of a formation from data recorded by an induction sonde in a borehole where well logging or other such operations are being performed.
The primary goal of induction logging is to obtain an accurate determination of the profile of the true conductivity (or its inverse the resistivity) of the earth formations surrounding the borehole. In particular, it is desired to obtain a high resolution reconstruction or image of the true formation resistivity (denoted by R.sub.t) profile including accurate resistivity values in thin beds (e.g. 2 foot-thick strata) which are frequently encountered in oil and gas wells. Modern induction tools, such as the tool described in U.S. Pat. No. 3,179,879 to Tanguy, have focussed multi-coil arrays and measure both the R-signal (in-phase) and X-signal (quadrature) components of the formation signal. It is also desirable to have at least two different radial depths of investigation including a deep array (ID) and a medium (IM) array. The arrays with different radial depths of investigation detect and correct for environmental effects such as the influence of the borehole and the invasion into the formation of fluids from the borehole. This requirement necessitates having arrays with relatively long transmitter-receiver spacings since borehole fluid invasion of sixty inches more is not uncommon. Thus, for example the ID array is designed to be able to see beyond the invaded zone and to have a signal representative of the virgin formation. Unfortunately, there is, with increasing depth of investigation in the radial direction, a loss of vertical resolution. That is, the instrumental induction tool vertical response functions have poor vertical resolution. The ID and IM vertical response functions in a homogeneous medium with infinite resistivity are known as geometrical factor or Doll response functions. In wells penetrating very low conductivity rock formations, the raw induction tool response for the ID and IM arrays, in the absence of invasion and neglecting the borehole, can be obtained by convolving the formation conductivity profile with the Doll response functions shown in FIG. 3. Ideally, it would be desirable for the tool vertical response functions to be very localized spatially (i.e., to be mathematically described by Dirac delta functions) about a central peak with no sidelobes so that the raw tool response would be the true formation conductivity profile. Another difficulty is that the induction tool response functions depend on the formation conductivity in a non-linear fashion. This has the effect that, even in homogeneous media, the vertical response functions change shape and spatial extent depending on the background conductivity. In a homogeneous medium, in the limit of formation conductivity approaching zero, the induction tool R-signal is proportional to the formation conductivity. As the formation conductivity increases, the R-signal increases less rapidly than the formation conductivity. This non-linear dependence on formation conductivity is known by those skilled in the art as skin effect.
Traditional induction log signal processing methods have exploited the approximate linearity of the induction tool response on formation conductivity by constructing linear inverse filters whose convolution with the tool response function produces a filtered response function which is more spatially localized (i.e., has a narrower central peak) and has reduced sidelobes. The spatial localization of the filtered response function provides better resolution whereas the reduced sidelobes suppress the shoulder effect. The shoulder bed effect generally occurs in resistive beds which are adjacent to more conductive beds. One of the purposes of inverse filters is to reduce the shoulder bed effect. Traditional induction log signal processing is based on simple inverse filters which have been used commercially in the well-logging industry for about forty years. The induction logs obtained by applying these filters to the raw measured log data are denoted by ILD (induction log deep) and ILM (induction log medium). The ILD curve is obtained by applying a three-point deconvolution filter and skin-effect boosting correction to the measured log data. The ILM curve simply involves a skin-effect boosting correction to the raw data. In FIG. 4, the ILD and ILM processed logs are illustrated. Note that the ILD and ILM curves show significant shoulder effects in all of the beds. The shoulder effect is often the main reason that the ILD and ILM curves do not read the true bed resistivities.
The maximum entropy method (MEM) has recently been used in many fields of science and engineering to obtain inversions of instrumentally blurred and noisy data. It has proved to be an especially powerful technique in image reconstruction and pattern recognition problems. It has also been used in exploration geophysics to process seismic data. It has not to data been commercially utilized in the well-logging industry. A recent article published by Dyos in "SPWLA transactions", 1987, applied the maximum entropy method to the inversion of R-signal data from the ID array. The inversions obtained by Dyos using this algorithm exhibited spurious oscillations at the blind frequencies of the ID array, as shown in FIG. 5. The present invention removes these spurious oscillations in the reconstructions obtained by Dyos.
More recent advances in induction log signal processing, such as the advances discussed in U.S. Pat. No. 4,471,436 issued to Schaefer et al., have developed filters which use the measured X-signals to improve the skin-effect correction. In spite of significant advances during the past decade in induction tool technology and inverse filter design, there are intrinsic limitations to the vertical resolution achievable without the risk of producing spurious artifacts and instabilities on the deconvolved conductivity profile.
Other approaches to induction log signal processing have used forward modeling and criteria such as least squares inversion to iteratively determine model parameters describing the formation conductivity profile. This approach is known as a parametric inversion because it assumes a specific model for the formation conductivity profile. A model with a step profile is often used. However, this approach has its limitations, that is, if the actual formation does not conform to the assumed model, the conductivity values determined from a parametric inversion can be very far from the truth. The maximum entropy method described in this application is not a parametric inversion; rather, it is similar to methods used in image reconstruction of instrumentally blurred and noisy data. The maximum entropy method attempts to extract all the information that can be safely extracted from the data. Its objective is to improve the resolution and accuracy of the estimated R.sub.t, but give results that are stable and reliable.